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Grete Hermann

This thread about famous women mathematicians on Cocktail Party Physics, reminded me of an interesting figure in history that I came across while doing researching for a Wikipedia article: Grete Hermann. (The Wikipedia article is a skeleton that I created; it could use a lot of work.)

Hermann was a student of Emmy Noether. Noether was one of the iconic figures of twentieth-century mathematics, a key figure in the century’s trend toward abstraction. A typical example is her proof of the Lasker-Noether theorem. The theorem, that every ideal has a primary decomposition, was originally proven for polynomial rings by Emanuel Lasker, using a difficult computational argument. Noether identified the key abstract condition behind the result — the ascending chain condition on ideals — and used it to give a shorter proof of a much more general theorem. Rings that satisfy the ascending chain condition on ideals are now known as Noetherian rings in her honor.

While Hermann was Noether’s student, her thesis was a throwback to the nineteenth century’s computational approach. Hermann showed that Lasker’s approach could be turned into an effective procedure for computing primary decompositions. Hermann did this before the invention of the computer, or even before the notion of an effective procedure had been formalized. (As her definition, Hermann used the existence of an explicit upper bound on time complexity, and gave such a bound for primary decomposition, and other questions in commutative ring theory.)

Hermann went on to work in philosophy and the foundations of physics. John Von Neumann had proposed a proof that a hidden variable theory of quantum mechanics could not exist. (A hidden variable theory is one that explains the random behavior of quantum mechanical systems in terms of unobserved deterministic variables.) Hermann discovered and published the flaw in Von Neumann’s proof back in 1935, a result that has no impact until it was rediscovered by John Bell some thirty years later.

(The thread on Cocktail Party Physics is instructive for just how unfamous mathematicians really are. For physicists, Karl Weierstrauss is an obscure historical figure. For mathematicians of course, Weierstrauss is five times as famous as Madonna and Britney Spears combined. It was interesting to learn that Sofia Kovalevskaya is not particularly well-known among physicists, even though part of her research was in classical mechanics.)